DVI and HDMI are technically the same. They kinda transfer digital signal the same way. The only difference is that the monitor must have an HDMI connection on its end to recognize HDMI audio and activate it. There is no such a thing as DVI audio for any monitor to recognize in the first place so the monitor will just provide video only by default since no device was ever programmed/designed to receive audio through DVI.

AMD/ATI cards starting Radeon HD 3000 series, if I'm not mistaking, were however programmed/designed to send audio through DVI as it is, once again, technically the same as HDMI, the part of which AMD took advantage!

I hope my clarification and English make sense

Edit:
You can read any reviews or specifications for video cards using the same chipset as the one you own (Radeon HD 3650), and find out it mentions that it provides HDMI audio though DVI adaptor!

Did you do that while the DVI cable in you original post was still connected?

If yes, you'll have to activate the other DVI port from the Catalyst Control Panel (multi-monitor setup) for the DVI-HDMI cable connected there to work. You can also completely disconnect the main DVI and replace it with that new DVI-HDMI cable while the computer is turned off.